数据、模型与决策(运筹学)课后习题和案例答案013

CHAPTER 13 FORECASTINGReview Questions13.1-1 Substantially underestimating demand is likely to lead to many lost sales, unhappy customers, and perhaps allowing the competition to gain the upper hand in the marketplace. Significantly overestimating the demand is very costly due to excessive inventory costs, forced price reductions, unneeded production or storage capacity, and lost opportunity to market more profitable goods. 13.1-2 A forecast of the demand for spare parts is needed to provide good maintenance service. 13.1-3 In cases where the yield of a production process is less than 100%, it is useful to forecast the production yield in order to determine an appropriate value of reject allowance and, consequently, the appropriate size of the production run. 13.1-4 Statistical models to forecast economic trends are commonly called econometric models. 13.1-5 Providing too few agents leads to unhappy customers, lost calls, and perhaps lost business. Too many agents cause excessive personnel costs. 13.2-1 The company mails catalogs to its customers and prospective customers several times per year, as well as publishing mini-catalogs in computer magazines. They then take orders for products over the phone at the company’s call center. 13.2-2 Customers who receive a busy signal or are on hold too long may not call back and business may be lost. If too many agents are on duty there may be idle time, which wastes money because of labor costs. 13.2-3 The manager of the call center is Lydia Weigelt. Her current major frustration is that each time she has used her procedure for setting staffing levels for the upcoming quarter, based on her forecast of the call volume, the forecast usually has turned out to be considerably off. 13.2-4 Assume that each quarter’s call volume will be the same as for the preceding quarter, except for adding 25% for quarter 4. 13.2-5 The average forecasting error is commonly called MAD, which stands for Mean Absolute Deviation. Its formula is MAD = (Sum of forecasting errors) / (Number of forecasts) 13.2-6 MSE is the mean square error. Its formula is (Sum of square of forecasting errors) / (Number of forecasts). 13.2-7 A time series is a series of observations over time of some quantity of interest. 13.3-1 In general, the seasonal factor for any period of a year measures how that period compares to the overall average for an entire year. 13.3-2 Seasonally adjusted call volume = (Actual call volume) / (Seasonal factor).13-1

13.3-3 Actual forecast = (Seasonal factor)(Seasonally adjusted forecast) 13.3-4 The last-value forecasting method sometimes is called the naive method because statisticians consider it naive to use just a sample size of one when additional relevant data are available. 13.3-5 Conditions affecting the CCW call volume were changing significantly over the past three years. 13.3-6 Rather than using old data that may no longer be relevant, this method averages the data for only the most recent periods. 13.3-7 This method modifies the moving-average method by placing the greatest weight on the last value in the time series and then progressively smaller weights on the older values. 13.3-8 A small value is appropriate if conditions are remaining relatively stable. A larger value is needed if significant changes in the conditions are occurring relatively frequently. 13.3-9 Forecast = α(Last Value) + (1 – α)(Last forecast). Estimated trend is added to this formula when using exponential smoothing with trend. 13.3-10 The one big factor that drives total sales up or down is whether there are any hot new products being offered. 13.4-1 CB Predictor uses the raw data to provide the best fit for all these inputs as well as the forecasts. 13.4-2 Each piece of data should have only a 5% chance of falling below the lower line and a 5% chance of rising above the upper line. 13.5-1 The next value that will occur in a time series is a random variable. 13.5-2 The goal of time series forecasting methods is to estimate the mean of the underlying probability distribution of the next value of the time series as closely as possible. 13.5-3 No, the probability distribution is not the same for every quarter. 13.5-4 Each of the forecasting methods, except for the last-value method, placed at least some weight on the observations from Year 1 to estimate the mean for each quarter in Year 2. These observations, however, provide a poor basis for estimating the mean of the Year 2 distribution. 13.5-5 A time series is said to be stable if its underlying probability distribution usually remains the same from one time period to the next. A time series is unstable if both frequent and sizable shifts in the distribution tend to occur. 13.5-6 Since sales drive call volume, the forecasting process should begin by forecasting sales. 13.5-7 The major components are the relatively stable market base of numerous small-niche products and each of a few major new products. 13.6-1 Causal forecasting obtains a forecast of the quantity of interest by relating it directly to one or more other quantities that drive the quantity of interest.

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13.6-2 The dependent variable is call volume and the independent variable is sales. 13.6-3 When doing causal forecasting with a single independent variable, linear regression involves approximating the relationship between the dependent variable and the independent variable by a straight line. 13.6-4 In general, the equation for the linear regression line has the form y = a + bx. If there is more than one independent variable, then this regression equation has a term, a constant times the variable, added on the right-hand side for each of these variables. 13.6-5 The procedure used to obtain a and b is called the method of least squares. 13.6-6 The new procedure gives a MAD value of only 120 compared with the old MAD value of 400 with the 25% rule. 13.7-1 Statistical forecasting methods cannot be used if no data are available, or if the data are not representative of current conditions. 13.7-2 Even when good data are available, some managers prefer a judgmental method instead of a formal statistical method. In many other cases, a combination of the two may be used. 13.7-3 The jury of executive opinion method involves a small group of high-level managers who pool their best judgment to collectively make a forecast rather than just the opinion of a single manager. 13.7-4 The sales force composite method begins with each salesperson providing an estimate of what sales will be in his or her region. 13.7-5 A consumer market survey is helpful for designing new products and then in developing the initial forecasts of their sales. It is also helpful for planning a marketing campaign. 13.7-6 The Delphi method normally is used only at the highest levels of a corporation or government to develop long-range forecasts of broad trends. 13.8-1 Generally speaking, judgmental forecasting methods are somewhat more widely used than statistical methods. 13.8-2 Among the judgmental methods, the most popular is a jury of executive opinion. Manager’s opinion is a close second. 13.8-3 The survey indicates that the moving-average method and linear regression are the most widely used statistical forecasting methods.

e) Considering the MAD values (5.2, 3.0, and 3.9, respectively), the averaging method is the best one to use. f) Considering the MSE values (30.6, 11.1, and 17.4, respectively), the averaging method is the best one to use. g) Unless there is reason to believe that sales will not continue to be relatively stable, the averaging method should be the most accurate in the future as well. 13.18 Using the template for exponential smoothing, with an initial estimate of 24, the following forecast errors were obtained for various values of the smoothing constant α: Smoothing Constant MAD MSE 0.1 2.7 9.4 0.2 2.8 10.2 0.3 3.0 11.2 0.4 3.1 12.4 0.5 3.3 13.8 Considering both MAD and MSE values, α = 0.1 is the best smoothing constant to use. 13.19 a) Answers will vary. Averaging or Moving Average appear to do a better job than Last Value. b) For Last Value, a change in April will only affect the May forecast. For Averaging, a change in April will affect all forecasts after April. For Moving Average, a change in April will affect the May, June, and July forecast. c) Answers will vary. Averaging or Moving Average appear to do a slightly better job than Last Value. d) Answers will vary. Averaging or Moving Average appear to do a slightly better job than Last Value.

Comparing MAD values (5.3, 10.0, and 8.1, respectively), the last-value method is the best to use of these three options. Comparing MSE values (36.2, 131.4, and 84.3, respectively), the last-value method is the best to use of these three options.

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c) Using the template for exponential smoothing, with an initial estimate of 120, the following forecast errors were obtained for various values of the smoothing constant α: Smoothing Constant MAD MSE 0.1 18.5 382.7 0.2 13.0 210.2 0.3 10.1 139.7 0.4 8.7 104.2 0.5 8.0 82.9 Considering both MAD and MSE, it appears that a high value for the smoothing constant is appropriate. d) Using the template for exponential smoothing with trend, using initial estimates of 120 for the average value and 10 for the trend, the following forecast errors were obtained for various values of the smoothing constants α and β : MAD MSE 0.1 0.1 25.4 919.6 0.1 0.3 21.2 634.1 0.1 0.5 17.7 450.6 0.3 0.1 13.5 261.9 0.3 0.3 9.8 144.1 0.3 0.5 8.8 111.5 0.5 0.1 8.4 116.1 0.5 0.3 7.0 72.2 0.5 0.5 6.5 61.1 Considering both MAD and MSE, it appears that a high value for both smoothing constants is appropriate. e) Management should use the last-value method to forecast sales. Using this method the forecast for January of the new year will be 166. Exponential smoothing with trend with high smoothing constants (e.g., α = 0.5 and β = 0.5) also works well. With this method, the forecast for January of the new year will be 165. 13.21 a) Shift in total sales may be due to the release of new products on top of a stable product base, as was seen in the CCW case study. b) Forecasting might be improved by breaking down total sales into stable and new products. Exponential smoothing with a relatively small smoothing constant can be used for the stable product base. Exponential smoothing with trend, with a relatively large smoothing constant, can be used for forecasting sales of each new product. c) Managerial judgment is needed to provide the initial estimate of anticipated sales in the first month for new products. In addition, a manger should check the exponential smoothing forecasts and make any adjustments that may be necessary based on knowledge of the marketplace.

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13.22 a) Answers will vary. Last value seems to do the best, with exponential smoothing with trend a close second. b) For last value, a change in April will only affect the May forecast. For averaging, a change in April will affect all forecasts after April. For moving average, a change in April will affect the May, June, and July forecast. For exponential smoothing, a change in April will affect all forecasts after April. For exponential smoothing with trend, a change in April will affect all forecasts after April. c) Answers will vary. last value or exponential smoothing seem to do better than the averaging or moving average. d) Answers will vary. last value or exponential smoothing seem to do better than the averaging or moving average. 13.23 a) Using the template for exponential smoothing, with an initial estimate of 50, the following MAD values were obtained for various values of the smoothing constant α: Smoothing Constant 0.1 0.2 0.3 0.4 0.5 Choose α = 0.1 b) Using the template for exponential smoothing, with an initial estimate of 50, the following MAD values were obtained for various values of the smoothing constant α: Smoothing Constant 0.1 0.2 0.3 0.4 0.5 Choose α = 0.2 MAD 1.69 1.66 1.71 1.82 1.93 MAD 1.49 1.58 1.67 1.76 1.86

13.25 a) Using the template for exponential smoothing with trend, with an initial estimates of 50 for the average and 2 for the trend and α = 0.2, the following MAD values were obtained for various values of the smoothing constant β : Smoothing Constant 0.1 0.2 0.3 0.4 0.5 Choose β = 0.1 MAD 0.74 0.75 0.76 0.77 0.78

b) Using the template for exponential smoothing with trend, with an initial estimates of 50 for the average and 2 for the trend and α = 0.2, the following MAD values were obtained for various values of the smoothing constant β : Smoothing Constant 0.1 0.2 0.3 0.4 0.5 Choose β = 0.1 MAD 2.61 2.76 2.87 2.99 3.05

For moving average, the forecast typically lie below the demands. For exponential smoothing, the forecasts typically lie below the demands. For exponential smoothing with trend, the forecasts are at about the same level as demand (perhaps slightly above). This would indicate that exponential smoothing with trend is the best method to use hereafter.

e) Moving average results in the best MAD value (13.30) and the best MSE value (249.09). f) Month January February March April May June July August September October November December MAD = 14.17 g) Moving average performed better than the average of all three so it should be used next year. Avg. Forecast341 345 375 400 451 497 459 537 369 354 677 592

Causal forecasting takes all the data into account, even the data from before changing conditions cause a shift. Exponential smoothing with trend adjusts to shifts in the underlying trend by placing more emphasis on the recent data.

f) An increase of 31 passengers can be attained. 13.39 a) If the sales change from 16 to 19 when the amount of advertising is 225, then the linear regression line shifts below this point (the line actually shifts up, but not as much as the data point has shifted up). b) If the sales change from 23 to 26 when the amount of advertising is 450, then the linear regression line shifts below this point (the line actually shifts up, but not as much as the data point has shifted up). c) If the sales change from 20 to 23 when the amount of advertising is 350, then the linear regression line shifts below this point (the line actually shifts up, but not as much as the data point has shifted up). 13.40 a) The number of flying hours is the independent variable and the number of wing flaps needed is the dependent variable. b)

Case13.1 a) We need to forecast the call volume for each day separately. 1) To obtain the seasonally adjusted call volume for the past 13 weeks, we first have to determine the seasonal factors. Because call volumes follow seasonal patterns within the week, we have to calculate a seasonal factor for Monday, Tuesday, Wednesday, Thursday, and Friday. We use the Template for Seasonal Factors. The 0 values for holidays should not factor into the average. Leaving them blank (rather than 0) accomplishes this. (A blank value does not factor into the AVERAGE function in Excel that is used to calculate the seasonal values.) Using this template (shown on the following page, the seasonal factors for Monday, Tuesday, Wednesday, Thursday, and Friday are 1.238, 1.131, 0.999, 0.850, and 0.762, respectively.

2) To forecast the call volume for the next week using the last-value forecasting method, we need to use the Last Value with Seasonality template. To forecast the next week, we need only start with the last Friday value since the Last Value method only looks at the previous day.A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 B C D E F G H I J K

The forecasted call volume for the next week is 5,045 calls: 1,254 calls are received on Monday, 1,148 calls are received on Tuesday, 1,012 calls are received on Wednesday, 860 calls are received on Thursday, and 771 calls are received on Friday.

The forecasted call volume for the next week is 4,712 calls: 1,171 calls are received on Monday, 1,071 calls are received on Tuesday, 945 calls are received on Wednesday, 804 calls are received on Thursday, and 721 calls are received on Friday.

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4) To forecast the call volume for the next week using the moving-average forecasting method, we need to use the Moving Averaging with Seasonality template. Since only the past 5 days are used in the forecast, we start with Monday of the last week to forecast through Friday of the next week.A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 B C D E F G H I J K

The forecasted call volume for the next week is 4,124 calls: 985 calls are received on Monday, 914 calls are received on Tuesday, 835 calls are received on Wednesday, 732 calls are received on Thursday, and 658 calls are received on Friday. 5) To forecast the call volume for the next week using the exponential smoothing forecasting method, we need to use the Exponential with Seasonality template. We start with the initial estimate of 1,125 calls (the average number of calls on non-holidays during the previous 13 weeks).

The forecasted call volume for the next week is 4,322 calls: 1,074 calls are received on Monday, 982 calls are received on Tuesday, 867 calls are received on Wednesday, 737 calls are received on Thursday, and 661 calls are received on Friday.13-43

b) To obtain the mean absolute deviation for each forecasting method, we simply need to subtract the true call volume from the forecasted call volume for each day in the sixth week. We then need to take the absolute value of the five differences. Finally, we need to take the average of these five absolute values to obtain the mean absolute deviation. 1) The spreadsheet for the calculation of the mean absolute deviation for the last-value forecasting method follows.A 1 2 3 4 5 6 7 8 9 B C D E F G H

This method is the least effective of the four methods because this method depends heavily upon the average seasonality factors. If the average seasonality factors are not the true seasonality factors for week 6, a large error will appear because the average seasonality factors are used to transform the Friday call volume in week 5 to forecasts for all call volumes in week 6. We calculated in part (a) that the call volume for Friday is 0.762 times lower than the overall average call volume. In week 6, however, the call volume for Friday is only 0.83 times lower than the average call volume over the week. Also, we calculated that the call volume for Monday is 1.34 times higher than the overall average call volume. In Week 6, however, the call volume for Monday is only 1.21 times higher than the average call volume over the week. These differences introduce a large error.

This method is the second-most effective of the four methods. Again, the reason lies in the average seasonality factors. Applying the average seasonality factors to an average call volume yields a much more accurate result than applying average seasonality factors to only one call volume. This method is not the most effective method, however, because the centralized call center experiences not only daily seasonality, but also weekly seasonality. For example, the call volumes in weeks 45 and 46 are much greater than the call volumes in week 6. Therefore, these larger call volumes inflate the average call volume, which in turn inflates the forecasts for Week 6. 3)The spreadsheet for the calculation of the mean absolute deviation for the movingaverage forecasting method appears below.A 1 2 3 4 5 6 7 8 9 B C D E F G H

This method is the most effective of the four methods because this method only uses the average week 5 call volume to forecast the call volumes for week 6. Again, applying the average seasonality factors to an average call volume yields a much more accurate result than applying average seasonality factors to only one call volume. Also, the average call volume used in this method is not overly inflated since it is an average of the week 5 call volumes, which are closer to the week 6 call volumes than any other of the 13 weeks.

This method is nearly as effective as the moving average. This method is a little more effective than the averaging forecasting method because the smoothing constant causes less weight to be placed on the call volumes in the earlier weeks. c) This problem is simply a linear regression problem. 1) To find a mathematical relationship, we use the Linear Regression template. The decentralized case volumes are the independent variables, and the centralized case volumes are the dependent variables. Substituting the case volume data, we obtain the following spreadsheet. The relationship is y = 1576 + 0.756x, where x is the decentralized case volume, and y is the estimated centralized case volume.A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 B C D E F G H I J

2) To forecast the week 6 call volume for the centralized call center, we simply input the week 6 decentralized case volume for the value of x in the Estimator section of the Linear Regression Spreadsheet (as shown in part 1 above). The value of y then represents the week 6 centralized case volume. We multiply this value of y by 1.5 to obtain the week 6 centralized call volume. Thus, the forecasted number of calls is 1.5 * 2,038.9 = 3,058. We then break this weekly call volume into daily call volume. We do this conversion by dividing the weekly call volume by the sum of the seasonal factors calculated in part (a) and then multiplying this weekly call volume by the appropriate seasonal factor to find the call volume for each of the five days of the week. The spreadsheet showing these calculations follows:A 1 2 3 4 5 6 7 8 9 10 B C

The forecasted call volume for week 6 is 3,046 calls: 757 calls are received on Monday, 692 calls are received on Tuesday, 611 calls are received on Wednesday, 520 calls are received on Thursday, and 466 calls are received on Friday.

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3) To calculate the mean absolute deviation, we need to subtract the true call volume from the forecasted call volume for each day in the sixth week. We then need to take the absolute value of the five differences. Finally, we need to take the average of these five absolute values to obtain the mean absolute deviation. The spreadsheet for the calculation of the mean absolute deviation follows.A 1 2 3 4 5 6 7 8 9 B C D E F G H

This forecasting method is by far the most effective method. The centralized center performs the same services and serves the same population as the decentralized center. Therefore, the call volume trends are the same. Once we have a factor to scale the decentralized call volumes to the centralized call volumes, we have a very effective forecasting method. d) We would definitely recommend using the causal forecasting method implemented in part (c) because it yields the lowest error. The causal method shows us that the call volume trends remain relatively the same year after year. We had to convert between case volumes and call volumes in part (c), however, and such a conversion introduces error. For example, what if a case generates a higher or lower number of calls? We therefore recommend that call volume data be meticulously recorded as the centralized center continues its operation. Once one year’s worth of call volumes have been collected, the causal forecasting model should be updated. The model should be updated to use the historical centralized call volume data instead of the historical decentralized case volume data.